In this section we outline the limits of our framework. As mentioned above, W-BNSL can be used to model Bayesian network learning under additional sparsity constraints like a bounded number of arcs. However, some important sparsity constraints are unlikely to be modeled with our framework, for example sparsity constraints that are posed on the moralized graph [1]}:
The moralized graph of a DAG \(D\)  is an undirected graph \(\mathcal {M}(D):=(V, E_1 \cup E_2)\)  with \(V:=N\) , \(E_1 := \lbrace \lbrace u,v\rbrace  \mid (u,v) \in A \rbrace \) , and \(E_2 := \lbrace \lbrace u,v\rbrace  \mid u \text{ and }v \text{ have a common child in }D \rbrace \) . The edges in \(E_2\)  are called moral edges. The NP-hard task of Bayesian inference can be solved more efficiently if the moralized graph of the network is treelike, that is, it has small treewidth [2]}. Thus, it is well motivated to learn a Bayesian network that satisfies a sparsity constraint that provides an upper bound on the treewidth of the moralized graph.
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